The mean value of |L(k,χ)|² at positive rational integers k ≥ 1

Stéphane Louboutin

Displaying similar documents to “The mean value of |L(k,χ)|² at positive rational integers k ≥ 1”

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , C being an absolute constant.

On discrete mean values of Dirichlet $L$ -functions

Ertan Elma (2021)

Czechoslovak Mathematical Journal

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Let $χ$ be a nonprincipal Dirichlet character modulo a prime number $p ⩾ 3$ and let $𝔞_{χ} : = \frac{1}{2} (1 - χ (- 1))$ . Define the mean value $ℳ_{p} (- s, χ) : = \frac{2}{p - 1} \sum ψ (mod p) ψ (- 1) = - 1 L (1, ψ) L (- s, χ \bar{ψ}) (σ : = ℜ s > 0) .$ We give an identity for $ℳ_{p} (- s, χ)$ which, in particular, shows that $ℳ_{p} (- s, χ) = L (1 - s, χ) + 𝔞_{χ} 2 p^{s} L (1, χ) ζ (- s) + o (1) (p \to \infty)$ for fixed $0 < σ < \frac{1}{2}$ and $| t : = ℑ s | = o (p^{(1 - 2 σ) / (3 + 2 σ)})$ .

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

A study of various results for a class of entire Dirichlet series with complex frequencies

Niraj Kumar, Garima Manocha (2018)

Mathematica Bohemica

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Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_{k} e^{〈 λ^{k}, z 〉}$ for which $(| λ^{k} {| / e)}^{| λ^{k} |} k! | a_{k} |$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

Some new sums related to D. H. Lehmer problem

Han Zhang, Wenpeng Zhang (2015)

Czechoslovak Mathematical Journal

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About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N (k; p)$ denote the number of all $1 \leq a_{i} \leq p - 1$ such that $a_{1} a_{2} \dots a_{k} \equiv 1 mod p$ and $2 ∣ a_{i} + {\bar{a}}_{i} + 1,$ $i = 1, 2, \dots, k$ . The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function $N (k; p),$ and give an interesting asymptotic formula...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

The Dirichlet-Bohr radius

Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)

Acta Arithmetica

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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_{n} n^{- s}$ we have $\sum_{n \leq x} | a_{n} | r^{Ω (n)} \leq s u p_{t \in ℝ} | \sum_{n \leq x} a_{n} n^{- i t} |$ . We prove that the asymptotically correct order of L(x) is ${(l o g x)}^{1 / 4} x^{- 1 / 8}$ . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...

Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura (2013)

Acta Arithmetica

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Let $p = 1 + 2^{e + 1} q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e + 1}$ (resp. order $d_{φ}$ dividing q), and let ψₙ be an even character of conductor $p^{n + 1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $K ₙ = ℚ (ζ_{(p - 1) p ⁿ})$ . It is well known that the Bernoulli number $B_{1, χ}$ is not zero, which is shown in an analytic way. In the extreme cases $d_{φ} = 1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T r_{n / 1} (ξ B_{1, χ}) \neq 0$ for any...

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $Γ = SL (2, ℤ)$ . Denote by $λ_{f} (n)$ the $n$ th normalized Fourier coefficient of $f$ . We are interested in the average behaviour of the sum $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{j} (a^{2} + b^{2})$ for $x \geq 1$ , where $a, b \in ℤ$ and $j \geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$ -functions and Rankin-Selberg $L$ -functions.

Estimates near the boundary for second order derivatives of solutions of the Dirichlet problem for the biharmonic equation

Vladimir A. Kondratiev, Olga A. Oleinik (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Per ogni soluzione della (1) nel dominio limitato $\Omega$ ,, appartenente a $H_{0}^{2}(\Omega)$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^{0}$ del contorno; si consente a $\partial\Omega$ di essere singolare in $x^{0}$ .

Zero-sums of length kq in $ℤ_{q}^{d}$

Silke Kubertin (2005)

Acta Arithmetica

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Waring's number for large subgroups of ℤₚ

Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)

Acta Arithmetica

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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t ₄ = p^{22 / 39 + ϵ}$ , $t ₅ = p^{15 / 29 + ϵ}$ and for s ≥ 6, $t ₛ = p^{(9 s + 45) / (29 s + 33) + ϵ}$ . For s ≥ 24 further improvements are made, such as $t_{32} = p^{5 / 16 + ϵ}$ and $t_{128} = p^{1 / 4}$ .

On sharp characters of type ${- 1, 0, 2}$

Alireza Abdollahi, Javad Bagherian, Mahdi Ebrahimi, Maryam Khatami, Zahra Shahbazi, Reza Sobhani (2022)

Czechoslovak Mathematical Journal

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For a complex character $χ$ of a finite group $G$ , it is known that the product $sh (χ) = \prod_{l \in L (χ)} (χ (1) - l)$ is a multiple of $| G |$ , where $L (χ)$ is the image of $χ$ on $G - {1}$ . The character $χ$ is said to be a sharp character of type $L$ if $L = L (χ)$ and $sh (χ) = | G |$ . If the principal character of $G$ is not an irreducible constituent of $χ$ , then the character $χ$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type ${- 1, 0, 2}$ . Here we prove that such a group with nontrivial...